A Multi Dimensional Assignment Formulation for New Product Development Problems

Abstract

This paper considers the application of an upper bounding technique to a maximization formulation of new product development (NPD) problems. NPD procedures are increasingly being used by many high-technology firms to rapidly develop multiple new products lines using a small but flexible workforce and infrastructure. Mathematically speaking, these problems are quite difficult and may be described as the allocation of heterogeneous resources to heterogeneous but perhaps interdependent activities. Typically, each resource may distribute its capacity among many activities, each resource is capable of processing more than one type of task to varying degrees of success, and activities may be processed by more than one resource either sequentially or simultaneously. NPDs may include precedence constraints, where sequencing, quite often in the form of simultaneity, for the beginning and ending of activities is carefully controlled while processing times and quality of services for resources are not independent. Network models for these problems are very difficult to pose since, unlike PERT, there are multiple projects, all interlinked with precedence. Consequently, a multi-dimensional assignment problem (MAP) formulation is proposed. Unfortunately, MAP data structures exhibit horrendous complexity for branch and bound when using continuous relaxations due to the very poor bound quality and cannot be solved for problems of even fairly small size. In this paper an upper bound problem is developed by exploiting a feature of the cost coefficients. This upper bound on the maximization formulation of the MAP with axial constraints preserves much of the structure of the original problem.